Precalculus Examples

$f (x) = - \ln (x + 1)$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = - \ln ((x + h) + 1)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = - \ln (x + h + 1)$

Step 2.1.2.2

The final answer is $- \ln (x + h + 1)$ .

$- \ln (x + h + 1)$

Step 2.2

Find the components of the definition.

$f (x + h) = - \ln (x + h + 1)$

$f (x) = - \ln (x + 1)$

$f (x + h) = - \ln (x + h + 1)$

$f (x) = - \ln (x + 1)$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \ln (x + h + 1) - (- \ln (x + 1))}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $- 1$ out of $- \ln (x + h + 1) - - \ln (x + 1)$ .

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Step 4.1.1.1

Move parentheses.

$\frac{- \ln (x + h + 1) - - 1 \ln (x + 1)}{h}$

Step 4.1.1.2

Factor $- 1$ out of $- \ln (x + h + 1)$ .

$\frac{- \ln (x + h + 1) - - 1 \ln (x + 1)}{h}$

Step 4.1.1.3

Factor $- 1$ out of $- - 1 \ln (x + 1)$ .

$\frac{- \ln (x + h + 1) - - \ln (x + 1)}{h}$

Step 4.1.1.4

Factor $- 1$ out of $- \ln (x + h + 1) - - \ln (x + 1)$ .

$\frac{- (\ln (x + h + 1) - \ln (x + 1))}{h}$

Step 4.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{- \ln (\frac{x + h + 1}{x + 1})}{h}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{\ln (\frac{x + h + 1}{x + 1})}{h}$

Step 5